Severi varieties and their varieties of reductions

Details Severi varieties and their varieties of reductions Severi varieties and their varieties of reductions

2003-06-23

arxiv.org/abs/math/0306328v1

Mathematics

Algebraic Geometry

41 pages

Scientific paper

We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension $3a$ and index $a+1$, for $a=1, 2, 4, 8$. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan's triality principle. We also prove that they are compactifications of affine spaces.

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